1. Mathematical Experiments

1.3 Universality

As was shown in the diagrams, subregions within the bifurcation diagram look
remarkably similar to each other and to the diagram as a whole. This self-similarity
was shown to repeat itself at ever finer resolutions. Such behavior is characteristic
of geometric entities called fractals (a topic I
will address in later chapters) and is quite common in iterated mappings. In the
period-doubling region, for instance, the whole region beginning at the first
bifurcation "λ1" [lambda one]
looks the same as either region beginning at the second bifurcation "λ2"
[lambda two] which looks the same as either region beginning at the third bifurcation
"λ3" [lambda three], and so on.
Interestingly enough, the distance between successive bifurcation points "λn"
[lambda n] shrinks geometrically in such a fashion that the ratio of the intervals approaches a constant value as "n" approaches infinity.

δ =

lim

n → ∞

λn − λn−1

[delta]

λn+1 − λn

This constant,
called Feigenbaum's number, crops up repeatedly in
self-similar figures and has an approximate value of…

Not only does Feigenbaum's constant reappear in other figures, but so do many
other characteristics of the bifurcation diagram. In fact, remarkably similar
diagrams can be generated from any smooth, one-dimensional, non-monotonic function
when mapped on to itself. A circle, ellipse, sine, or any other function with
a local maximum will produce a bifurcation diagram with period-doublings whose
ratios approach "δ" [delta]. Together with
a second constant "α" [alpha], the scaling
factor "δ" [delta] demonstrates a universality
previously unknown in mathematics: metrical universality.
The behavior of the quadratic map is typical for many dynamical systems. One year
after their discovery, the period-doubling route to chaos and the constants "α"
[alpha] and "δ" [delta] appeared in an unruly
mess of equations used to describe hydrodynamic flow. This might not be so amazing
if it weren't for the fact that Feigenbaum's constants were originally derived
from a mathematical model of animal populations. In the segmented, fragmented
world of modern science hydrodynamicists and population biologists rarely interact
with one another. The realization that a set of five coupled differential equations
describing turbulence could exhibit the same fundamental behavior as the one-dimensional
map of the parabola on to itself was a key event in the history of mathematics.

This chapter has been devoted to the exploration of the simple one-dimensional iterative mapping

ƒ: x → x2 + c

where "x" and "c" were real numbers. The statement was
made that the behavior of this system is typical for "any smooth, one-dimensional,
non-monotonic function when mapped on to itself." Many books on chaos mention
how Feigenbaum's constants and the period-doubling route to chaos appear in other
one-dimensional mappings, but few provide examples. Thus, I felt it necessary
to explore the behavior of additional functions to see if such results were really
universal. The results were rather interesting and quite unexpected. Let's look
at the bifurcation diagrams for some other mappings.

Bifurcation diagramƒ: x → c sin x

The structure of the bifurcation diagram is similar to that of ƒ: x → x2 + c
with a period-doubling path to chaos and an ergodic region with odd-period windows.
In the quadratic case, the bifurcation diagram was finite and ended at a parameter
value beyond which all orbits escaped to infinity. In the sinusoidal case, however,
the map continues with an abrupt bump in the chaotic regime. This region is punctuated
with even-period windows; the most prominent being a four-cycle. The chaotic
regime widens and then terminates on a two-cycle, each side of which bifurcates.
Again, the ergodic region bumps out and the pattern repeats itself ad infinitum
as shown below.

Bifurcation diagram on a larger intervalƒ: x → c sin x

As a variation on a theme, I tried the mapping ƒ: x → sin
x + c. This is more like the quadratic mapping in that the parameter
raises and lowers the function without changing its shape. As expected, the orbits
gave a bifurcation diagram nearly identical to that for the quadratic map, but
with a bit of a twist.

Bifurcation diagramƒ: x → sin (π x) + c

Again, we see an abrupt change from behavior characteristic of the quadratic
map to a broad ergodic region. As with the previous sinusoidal map, this extra
ergodic region has windows of even periodicity. (Note how the largest windows
are open on one side.) This region ends with another quadratic-like bifurcation
diagram rotated 180° from the first. The behavior of the diagram on the interval
[2, 4] is identical to that on [0, 2] only shifted two units higher.
Thus, the interval [0, 2] is characteristic of the remainder of the parameter
values and can be used as a unit cell. The full diagram runs diagonally through
the origin across the parameter space from negative infinity to positive infinity

Some more mundane bifurcation diagrams are shown below. All functions are smooth
and non-monotonic. Note that each one undergoes a period-doubling route to chaos
and that there are always windows of odd periodicity within the chaotic regime.

Some more bifurcation diagrams

x → cx (1 − x2)

x → cx3 (1 − x)

x → c (1 − (2x − 1)4)

x → cx (1 − x)

How about a non-monotonic function that is not smooth? The function below is called the tent function for obvious reasons.

The tent functionƒ: x → c (1 − 2 | x − ½ | )

For parameter values in the interval [0, ½] all well-behaved orbits collapse to zero while parameters greater than
one drive all orbits to infinity. The first diagram below shows the orbits
over the range [½, 1] where the behavior can be called "interesting".
While there is a superficial similarity to many of the previous mappings
(the two ergodic "arms" merging into one for example) this one does not exhibit period-doubling
or windows. What look like single lines turn out to be, on closer examination,
pairs of lines. When these are examined in more detail, they also turn
out to be pairs of lines and so on. Orbits in the non-ergodic regions are
not periodic but tend to cluster together and thus appear to have even
periodicity. There is no bifurcation with this map. Those orbits that are
neither stable nor ergodic most likely form a Cantor
set. I have yet to find any mention of the surprisingly odd behavior of this
map in the popular literature.

The first chapter introduces the basics of one-dimensional iterated maps. Take a function y = ƒ(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970s, a whole new branch of mathematics arose from the simple experiments described in this chapter.

The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them.

The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension.

The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos.